I take it that there are two ways of interpreting your objection to the use of formal systems. One is a ban on the significance of formal systems tout court---they are merely a game. The other is a ban on formal systems as tools for interpreting ordinary practices. — Nagase
Neither really, just that (as you later allude to) as models they need to have something to hook them back to the thing they're modelling. In models of physics, for example, that's experimental hypothesis testing. I have no objection to models in principle (I'm pretty much a model-dependent realist so models are my building blocks of reality - quite important). I also have a great deal of sympathy for the work of Nancy Cartwright as a consequence. It's just that logical models of things like truth make what, to me, is a massive assumption about the way language works in psychology which is largely unsupported by the evidence in that field.
When we assign a truth value to a proposition, the assumption goes, we're performing some analysis of the syntax, the semantics of the actual proposition and the result is some binary value (true/false). But this is not what we see happening in the brain, nor do we infer it from behavioural experiments.
When reading a sentence with a semantic mismatch "my dog is house" fro example, there are N400 elevations associated with language comprehension which do not trigger responses in higher cortices. We dismiss, or flag such sentences as being 'untrue' without recourse to any logical processing whatsoever. that a dog can't be a house i just part of what learning how to use the words 'dog' and 'house' entail, it's has nothing to do with the logical truth value of the proposition - these processes are virtually identical to those seen with syntactic mismatches "my dog is quietly".
Semantic mismatches produce the same neural responses here regardless of the truth of the proposition being assessed.
There was a key study done in Germany a few years ago which presented subjects with opposing groups of sentences to assess - true/matched "Africa is a continent"; true/mismatched "Saturn is not a continent"; false matched "Saturn is not a planet" and false/mismatched "Africa is not a planet". What they found was that the truth evaluation method used was context dependent. Those tasked with evaluation (rather than sorting) engaged a different process despite earlier experiments showing the n400 response to sematic mismatching.
Basically sentences which are meaningless by lack of sematic matching are processed as such prior to, and independent of their truth evaluation. "This sentence is false" doesn't strike us a odd because we don't know how to evaluate the truth condition. It strikes us as odd because there's a semantic mismatch (sentences alone aren't the sorts of things that can be false).
Truth evaluation (as opposed to sematic mismatch assessment) is a complex process. Fluent sentences (presented in say a clear, high contrast font) are more likely to be assessed as true, even by experts in the subject of the sentence, than ones in a low contrast font. Even at expert, highly specific knowledge assessment, the linguistic aspects of the sentence (fluency, grammar, prosody, source...) play a part in truth evaluation.
Finally, even when we arrive at a fairly uncomplicated truth-evaluation outside of pure linguistic comprehension, we find that semantic processing is embodied almost entirely. Sensory-motor, auditory, visual, olfactory, etc... are engaged in the processing of state evaluation in concepts pertaining to those relevant cortices. "this sentence is true/false" would not be processed in the same way as ""my dog is green" is true/false" because the very processing mechanism for the truth evaluation of "my dog is green" relies on the visual cortex's model of 'dog' and 'green' not as logical concepts but as exterior world states to which there is an appropriate response.
Basically assessing the truth value of sentences is a context dependent involving syntactic and semantic assessments, socially mediated source judgements, emotional valence, and embodied response rehearsal. We're basically classifying these propositions on the basis of what we'd
do about them, not on the basis of their logical coherence.
Logical models may well be a very positive tool in some areas, I'm not trying to dismiss their utility entirely, but when dealing with something like our mental processing, they have to be indexed to what's actually happening.